Let S be the sum of the integers 25-210:
S = 25 + 26 + 27 + … + 208 + 209 + 210
Let S₃, S₄, and S₁₂ denote the sums of the integers in S that are multiples of 3, 4, or 12, respectively. We'll also count how many terms each sum involves; it'll be useful later.
S₃ = 27 + 30 + 33 + … + 204 + 207 + 210 … … … (62 terms)
S₃ = 3 (9 + 10 + 11 + … + 68 + 69 + 70)
S₄ = 28 + 32 + 36 + … + 200 + 204 + 208 … … … (46 terms)
S₄ = 4 (7 + 8 + 9 + … + 50 + 51 + 52)
S₁₂ = 36 + 48 + 60 + … + 180 + 192 + 204 … … … (15 terms)
S₁₂ = 12 (3 + 4 + 5 + … + 15 + 16 + 17)
Let's look at S₃ :
S₃ = 3 (9 + 10 + 11 + … + 68 + 69 + 70)
By reversing the order of the sum, we get
S₃* = 3 (70 + 69 + 68 + … + 11 + 10 + 9)
Of course S₃ = S₃*, I'm just calling it something else temporarily. Notice that every term in the same position of either sum adds up the same number.
9 + 70 = 79
10 + 69 = 79
11 + 68 = 79
and so on. Then
S₃ + S₃* = 3 (79 + 79 + 79 + … + 79 + 79 + 79)
or
2S₃ = 3 × 62 × 79 ==> S₃ = 7,347
We can compute the other two sums in the same way.
S₄ = 4 (7 + 8 + 9 + … + 50 + 51 + 52)
S₄* = 4 (52 + 51 + 50 + … + 9 + 8 + 7)
==> 2S₄ = 4 × 46 × 59 ==> S₄ = 5,428
S₁₂ = 12 (3 + 4 + 5 + … + 15 + 16 + 17)
S₁₂* = 12 (17 + 16 +15 + … + 5 + 4 + 3)
==> 2S₁₂ = 12 × 15 × 20 ==> S₁₂ = 1,800
Then the sum you want is
S₃ + S₄ - S₁₂ = 10,975
We subtract S₁₂ because each of its terms is counted twice (once in S₃ and again in S₄).
A map in a particular city shows that a city block to be a rectangle that is 264 feet wide and 900 feet long. If a person was to walk from the top left corner to the lower right corner along the diagonal of the block, they would walk approximately _____. Round your answer to the nearest whole foot. Enter only the number.
Answer:
1164
Step-by-step explanation:
The person would walk approximately 938 feet from the top left corner to the lower right corner of the city block.
What is Pythagoras's theorem?Pythagoras's theorem states that in a right-angled triangle, the square of one side is equal to the sum of the squares of the other two sides.
As per the given figure, the length of the hypotenuse is the distance the person would walk, and the lengths of the other two sides are 264 feet and 900 feet.
Using the formula for the Pythagorean theorem, we can set up the following equation:
c² = a² + b²
where c is the length of the hypotenuse, a is the length of one of the other sides, and b is the length of the remaining side.
In this case, we can substitute the known values for a, b, and c to get:
c² = 264² + 900²
c² = 69696 + 810000
c² = 879696
c = √879696
c = 937.92110
Rounded to the nearest whole foot.
c = 938 feet
Thus, the person would walk approximately 938 feet from the top left corner to the lower right corner of the city block.
Learn more about Pythagoras's theorem here:
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There are 35 times as many students at Wow University as teachers. When all the students and teachers are seated in the 8544 seat auditorium, 12 seats are empty. How many students attend Wow University.
A. 237
B. 249
C. 8295
D. 8124
Answer:
C. 8296
Step-by-step explanation:
Answer:
c.8295
Step-by-step explanation:
8544-12 = 8532
8532-237 =8295
The dot plot below represents this set of data:
30, 29, 30, 28, 28, 27, 22, 28, 28, 22, 27, 26, 25, 27
Answer:
True
Step-by-step explanation:
All of them match up
The midpoint of a sègment is (6,-4) and one endpoint is (13,-2). Find the coordinates of the other endpoint.
let other one be (x,y)
We know midpoint formula
[tex]\boxed{\sf (x,y)=\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)}[/tex]
[tex]\\ \sf\longmapsto (6,-4)=\left(\dfrac{13+x}{2},\dfrac{-2+y}{2}\right)[/tex]
[tex]\\ \sf\longmapsto \dfrac{13+x}{2}=6[/tex]
[tex]\\ \sf\longmapsto 13+x=12[/tex]
[tex]\\ \sf\longmapsto x=12-13[/tex]
[tex]\\ \sf\longmapsto x=-1[/tex]
And
[tex]\\ \sf\longmapsto \dfrac{-2+y}{2}=-4[/tex]
[tex]\\ \sf\longmapsto -2+y=-8[/tex]
[tex]\\ \sf\longmapsto y=-8+2[/tex]
[tex]\\ \sf\longmapsto y=-6[/tex]
Answer:
(- 1, - 6 )
Step-by-step explanation:
Given endpoints (x₁, y₁ ) and (x₂, y₂ ) then the midpoint is
( [tex]\frac{x_{1}+x_{2} }{2}[/tex] , [tex]\frac{y_{1}+y_{2} }{2}[/tex] ) ← midpoint formula
Use this formula on the endpoints and equate to the coordinates of the midpoint.
let the other endpoint = (x, y) , then
[tex]\frac{13+x}{2}[/tex] = 6 ( multiply both sides by 2 )
13 + x = 12 ( subtract 13 from both sides )
x = - 1
[tex]\frac{-2+y}{2}[/tex] = - 4 ( multiply both sides by 2 )
- 2 + y = - 8 ( add 2 to both sides )
y = - 6
The coordinates of the other endpoint are (- 1, - 6 )
Help anyone can help me do this question,I will mark brainlest.
Answer:
Hello,
Step-by-step explanation:
Loi d'Ohm: U=R*I
a) U=20*0.5=10 (amp)
b) I=U/R=50/0.5=100 (amp)
An arc length is a fractional part of the
circumference of a circle. The area of a
sector is a fractional part of the area of a
circle
The stained glass circle- head
the window has a 2 -inch wide
frame. The grills divide the
semicircular glass plane into
four congruent regions
Using detailed steps, describe your
solution to the problems below
Your steps should be clear enough so that
any geometry student can complete
them
A. Find the area of the blue region
B. Find the perimeter of the outer window
frame
Answer:
Each of the 4 sectors have an area:
πR²/8 - πr²/8, whereR = 28/2 - 2 = 12 inr = 6/2 = 3 inFind the area:
A = π/8(12² - 3² ≈ 53 in²Outer perimeter of the frame:
P = d + πd/2 =28( 1 + π/2) ≈ 72 in*Write a linear equation that goes through the points (-2,6) and (1,1).
Answer:
Equation of the line is 3y = -5x + 8
Step-by-step explanation:
Slope:
[tex]{ \bf{m = \frac{y _{2} - y _{1} }{x _{2} - x _{1} } }}[/tex]
substitute:
[tex]{ \sf{m = \frac{1 - 6}{1 - ( - 2)} }} \\ { \sf{m = - \frac{5}{3} }}[/tex]
General equation of a line:
[tex]{ \boxed{ \pmb{y = mx + b}}}[/tex]
b is the y-intercept.
Consinder point (1, 1):
[tex]{ \sf{1 = ( - \frac{5}{3} \times 1) + b}} \\ { \sf{b = \frac{8}{3} }}[/tex]
Substitute:
[tex]{ \sf{y = - \frac{5}{3} x + \frac{8}{3} }} \\ \\ { \sf{3y = - 5x + 8}}[/tex]
can you help me with this please.
Answer:
98 is the answer
Step-by-step explanation:
you add 8 each time
Answer:
a₁₃ = 98
Step-by-step explanation:
The nth term of an arithmetic sequence is
[tex]a_{n}[/tex] = a₁ + (n - 1)d
where a₁ is the first term and d the common difference
Here a₁ = 2 and d = a₂ - a₁ = 10 - 2 = 8 , then
a₁₃ = 2 + (12 × 8) = 2 + 96 = 98
Mar-Vell Studios' blockbuster movie Gray Widow: It's About Time She Got a Movie premiered last weekend simultaneously in theaters and on streaming video. On opening weekend, the total revenue from the movie was $127,265,468, with the movie making 75% more revenue in theaters than it did via streaming video. How much money did Gray Widow make in theaters on opening weekend
Answer:
Gray Widow made $111,357,284.5 in theaters on opening weekend.
Step-by-step explanation:
75% more revenue in theaters than it did via streaming video.
The sum of the revenues is 100%.
So the theaters were responsible for the following percentage:
[tex]P = 50 + \frac{75}{2} = 50 + 37.5 = 87.5[/tex]
How much money did Gray Widow make in theaters on opening weekend?
87.5% of the total revenue, that is, 0.875 out of $127,265,468.
0.875*127,265,468 = $111,357,284.5
Gray Widow made $111,357,284.5 in theaters on opening weekend.
2+ (-3)^2 - (-1)
please explain! :3
Answer:
12
Step-by-step explanation:
Hello!
First of all you have to understand the order of operations.
PEMDAS
Parenthesis, Exponent, Multiplication/Divison, Addition, Subtraction.
From this you can see that we first have to deal with the exponent, then multiplication, then the addition.
So we get 2+9-(-1)
The 9 is from -3^2
2+9+1
11+1
12.
Hope this helps!
PLEASE HELP
Simplify by combining like terms.
2/3b - 2 3/4b =??
Step-by-step explanation:
the Anwer is 2-6b(3/4b) / 3b
Answer: -2[tex]\frac{1}{12}[/tex] b or -[tex]\frac{25}{12}[/tex] b
Step-by-step explanation:
[tex]\frac{2}{3}[/tex] b - 2 [tex]\frac{3}{4}[/tex] b
= [tex]\frac{2}{3}[/tex] b - [tex]\frac{11}{4}[/tex] b
= [tex]\frac{8-33}{12}[/tex] b
= -[tex]\frac{25}{12}[/tex] b
= -2[tex]\frac{1}{12}[/tex] b
Om had a balance of 546 in his checking account. He made a deposit of $56, and wrote checks for $17 and $30. His bank charged him a $3 service fee. How much
honey does Om have in the account?
Answer:
540
Step-by-step explanation:
546-56=490
490+17+30=537
537+3=540
frank bought 3 action figures 2 board games and 3 toy cars at the toy store . The action figure coast 1.95$ each, the board games coast 5.10$ a piece and the toy cars were 1.00$ if he paid with a twenty dollar bill, how much change should he get back?
Answer:
-5
Step-by-step explanation:
PLEASE HELP!
Find an equation in standard form for the ellipse with the vertical major axis of length 16 and minor axis of length 10.
Refer the attached image for the answer
HOPE SO IT HELPS YOU
pls somebody help me wth this
Answer:
8 puzzles, and 840 pieces
Step-by-step explanation:
420 / 4 = 105
630 / 6 = 105
Option 2 is the correct answer,
Reason
840 / 8 = 105
A. What are the approximate coordinates of the point of intersection of the two
graphs? What does this mean in terms of the two mold populations?
B. The area of mold A is given by the function Ad) = 100 -0.25 .When will this
mold cover 1000 square millimeters? Explain your reasoning.
The two mold populations are equal after 4.5 years and the number of days is 9.2
Calculating the approximate coordinates of intersectionFrom the question, we have the following parameters that can be used in our computation:
The graph
On the graph, we have the point of intersection to be
(x, y) = (4.5, 3)
The means that the two mold populations are equal after 4.5 years
Estimating when the moid will cover 1000 sq mlGiven that
A(d) = 100e⁰.²⁵ᵇ
We have
100e⁰.²⁵ᵇ = 1000
Divide by 100
e⁰.²⁵ᵇ = 10
So, we have
0.25d = ln(10)
Evaluate
0.25d = 2.3
Divide
d = 9.2
Hence, the number of days is 9.2
Read more about exponential functions at
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Martina is recording the percentages she earned on each quiz in her math class. Here are her results for the last 7 quizzes.
74, 64, 90, 74, 92, 68, 71
Find the range of the data set.
=___
Answer:
28
Step-by-step explanation:
Subtract highest number from lowest.
92-64 = 28
100 points please help... i think the answer is 288 pie squared but not sure
What is the surface area of a dome (a half sphere) with a radius of 12 meters?
576 pie meters squared
48 pie meters squared
288 pie meters squared
96 pie meters squared
9514 1404 393
Answer:
(c) 288π m²
Step-by-step explanation:
The area of a sphere is found using the formula ...
A = 4πr²
Then the area of a half-sphere with radius 12 m will be ...
A = (1/2)(4π)(12 m)² = 288π m²
PLEASE HELPPP
Below is a function. Create an input/output table for the function. Choose 4 different inputs and complete the table.
y=2x-1
Answer:
y x
-1 0
1 1
3 2
5 3
This is a table that follows the above function
[tex]integrate \: ln(x) [/tex]
Answer:
ln(x) = 1/x... It's a basic rule of Calculus.
Mary wants to get spray foam insulation in her attic space. Shown here is a diagram of her attic - how much spray foam does she need to buy
Answer:
455
Step-by-step explanation:
(1/2(10)(70))13
Mary needs to buy [tex]455~m^3[/tex] spray foam.
The correct answer is an option (B)
What is the volume of the triangular prism?"Volume = base area × length,
where, Base area = area of triangle
Length = length(or height) of the triangular prism"
For given question,
Mary wants to get spray foam insulation in her attic space.
To find the amount of spray foam she needs to buy, we need to find the volume of the given triangular prism.
First we find the base area of the given triangular prism.
For a triangular base,
base = 7 m and height = 10 m
Using the formula of the area of the triangle,
[tex]A=\frac{1}{2}\times 7\times 10\\\\ A=7\times 5\\\\A=35~sq.m.[/tex]
Now, we find the volume of the triangular prism.
[tex]V=A\times l\\\\V=35\times 13\\\\V=455~m^3[/tex]
Therefore, she needs to buy [tex]455~m^3[/tex] spray foam.
The correct answer is an option (B)
Learn more about the volume of triangular prism here:
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Find the range of the data set. Round to the nearest tenth.
Range = Highest - Lowest
Range = 23,950 - 6,950
Range = 17,000
Must click thanks and mark brainliest
Answer:
The range is 17000.
Step-by-step explanation:
How you find range is by finding the difference between the maximum and minimum number from the data set.
So, the maximum number is 23950 and minimum is 6950 from the data set.
23950 - 6950 = 17000.
Convert 2 5/8 to a decimal
Answer:
2.625
Step-by-step explanation:
First convert mixed fraction to improper fraction,
=> 21/8
Then by normal division you will get,
=> 2.625
Please help me solve this quickly! Will give brainliest!
Answer:
y = 5[tex]\sqrt{6}[/tex]
Step-by-step explanation:
JLT forms the special 90-60-30 right triangle and the side length for the given measure follows as 2a-a[tex]\sqrt{3}[/tex]-a
since the side length that sees 90 degrees is 10 then x = 5
according to Euclidian theorem :
y^2 = LK*KI (LK = 15 -x, KI = 15)
y^2 = 10*15
y^2 = 150 find the root for both sides
y = 5[tex]\sqrt{6}[/tex]
The dimension of a rectangle are given in base 2.The width is 10 base 2 cm and the length 11 base 2 cm.Find the perimeter
Answer:
2l+2w
2 (11) +2 (10)
22+20
42
what is the ratio of the surface area to the volume of a cube with sides length 5
Answer:
6 : 5
Step-by-step explanation:
Volume of a cube = s^3
5^3 = 125
Surface area of a cube = 6s^2
6(5^2) = 6(25) = 150
150 : 125
30 : 25
6 : 5
2 1/2 + 3 1/4 = ? pls
Answer:
5 [tex]\frac{3}{4}[/tex]
Step-by-step explanation:
2 [tex]\frac{1}{2}[/tex] + 3 [tex]\frac{1}{4}[/tex]
= (2 + 3) + [tex]\frac{1}{2} + \frac{1}{4}[/tex] Here we add the two whole numbers first
= 5 + [tex]\frac{1}{2} + \frac{1}{4}[/tex]
= 5 + [tex]\frac{2}{4} +\frac{1}{4}[/tex] Here you bring the two fractions to a common denominator and to to this you find the highest common factor of the denominators.
= 5 + [tex]\frac{3}{4}[/tex] Here we added 1/4 and 2/4
= 5 [tex]\frac{3}{4}[/tex] This is the answer simply add 5 and 3/4
HOPE THIS HELPED
Answer:
Step-by-step explanation:
Your answer is 5 3/4
use trigonometry to find the unknown side (round to 1 decimal)
Answer:
a = 7.5
Step-by-step explanation:
Using the sine ratio in the right triangle and the exact value
sin30° = [tex]\frac{1}{2}[/tex] , then
sin30° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{a}{15}[/tex] = [tex]\frac{1}{2}[/tex] ( cross- multiply )
2a = 15 ( divide both sides by 2 )
a = 7.5
A rectangle with an area of 3990 cm2 is x centimeters wide and (x+4) centimeters long. To the nearest tenth of a centimeter, the width and length are
Answer: width:60.2 cm
Length: 64.2 cm
Step-by-step explanation:
Given
The area of the rectangle is [tex]3990\ cm^2[/tex]
Width of the rectangle is [tex]x\ cm[/tex]
Length of the rectangle is [tex]x+4\ cm[/tex]
Area of the rectangle is the product of length and width
[tex]\therefore 3990=(x+4)x\\\Rightarrow 3990=x^2+4x\\\Rightarrow x^2+4x-3990=0\\\Rightarrow x=60.198\ or\ -65.198\ cm\\\text{Neglecting negative term}\\\Rightarrow x=60.198\approx 60.2\ cm[/tex]
Width of the rectangle is [tex]60.2\ cm[/tex]
Length of the rectangle is [tex]60.2+4=64.2\ cm[/tex]
Use the figure to find the Total Area.
Answer:
60
Step-by-step explanation:
Multiply length times width times height
Answer:
C or 104
Step-by-step explanation:
I assume this is surface area, and I don't know what kinda sadist person would make everyone learn in "Total Area," but it is what it is.
So, to find total area, we need to literally multiply every different number with the other.
(You should not do it the way I did, as I did it in the easiest and simpliest way to understand possible.)
Actually, you get the length, width, and height of a figure and do what is below pretty much.
You multiply the length by the width, the length by the height, and the width by the height (doesn't have to be in this order, this is just to make it easy to remember which ones you did.)
We multiply the
5 * 6
5 * 2
and then the
6 * 2
This accounts for every possible case, as multiplying 2 * 5 gives the same thing as the second expression anyways.
So we then solve these three expressions.
5 * 6 = 30
5 * 2 = 10
and then the
6 * 2 = 12
And we add them to get
30 + 10 + 12
= 52.
Then we multiply by 2, because there are two cases of each face of the shape.
You need to remember that there is 3 other faces on the shape you are looking at, it's just that the shape/figure is facing you in a way that you can't see them.
After multiplying by 2, we get
52 * 2
= 104, which is C.
TAKE A SCREENSHOT NOT A PICTURE AS WE DON'T NEED TO SEE SOME 420P MONITOR FROM 1987 THAT SHOWS EVERY PIXEL PLEASE AND THANK YOU :D
